Lattice-ordered group

Summary

A lattice-ordered group, or $\ell$-group, is an ordered group whose order is a lattice. Any two elements have a least upper bound and a greatest lower bound, and multiplication preserves the order on both sides. Every linearly ordered group is an $\ell$-group, and so are groups of real-valued functions with the pointwise order. In an $\ell$-group each element splits into a positive part and a negative part, which gives an absolute value and a rich arithmetic of inequalities.